   Chapter 12.2, Problem 46E

Chapter
Section
Textbook Problem

# Suppose that a and b are nonzero vectors that are not parallel and c is any vector in the plane determined by a and b. Give a geometric argument to show that c can be written as c = sa + tb for suitable scalars s and t. Then give an argument using components.

To determine

To give: A geometric argument to show the vector c=sa+tb.

Explanation

Given:

Non zero and non parallel two-dimensional vectors a and b and any vector c.

Definition:

Scalar multiplication:

Consider a scalar c and a vector v. The scalar multiple cv, which is a vector with a length more than |c| times of vector v in same direction of v.

Consider two-dimensional vectors a=a1,a2, b=b1,b2, and c=c1,c2. The vectors are located in xy-plane as they are two-dimensional. Sketch the vectors a, b, and c in xy-plane from origin. Name the origin as O. Extend the vector a using line A and vector b using B.

Draw the parallel lines A and B to vectors a and b from terminal point of vector c. Name the intersecting point of lines A and A as P and intersecting point of B and B as Q.

From explanation, sketch of vectors is shown in Figure 1.

From Figure 1, write the expression for vector c.

c=OP+OQ

From Figure 1, write the expression for scalar s by the use of definition.

s=|OP||a|

Here,

|OP| is positive length of OP,

|a| is positive length of vector a.

From Figure 1, write the expression for scalar t by the use of definition.

t=|OQ||b|

Here,

|OQ| is positive length of OQ,

|b| is positive length of vector b.

From Figure 1, write the expression for vector c in terms of vectors a and b.

c=sa+tb (1)

The x and y-coordinates of vector c are estimated by the use of equation (1). Substitution of x-coordinated of vectors a and b along with scalar results x-coordinate of vector c and substitution of y-coordinates of vectors a and b along with scalar provides the value of y-coordinate of vector c.

Substitute a1 for a, b1 for b, and c1 for c in equation (1),

c1=sa1+tb1 (2)

Multiply a2 on both sides of equation (2).

c1a2=a2(sa1+tb1)

c1a2=sa1a2+tb1a2 (3)

Substitute a2 for a, b2 for b, and c2 for c in equation (1),

c2=sa2+tb2 (4)

Multiply a1 on both sides of equation (4).

a1c2=a1(sa2+tb2)

c2a1=sa1a2+tb2a1 (5)

Subtract equation (3) from (5)

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